Bi-annual Algebraic and Tropical Meetings of
Brown and YaLE
(BATMOBYLE)

Spring 2018 @ Brown
May 1, 2018


The BATMOBYLE is a vehicle for bringing together the algebraic and tropical geometry groups of Brown and Yale (and surrounding universities!) for a biannual day of talks.

Abstracts
 Isabel Vogt (MIT) -- Interpolation problems for curves in projective space

In this talk we will consider the following problem: when does there exist a curve of degree d and genus g in Pr though n general points?
Asher Auel (Yale) -- The period-index problem for surfaces over a local field

Determining bounds for the index of a Brauer class in terms of its period is a historically significant problem in pure algebra that has seen recent progress with the introduction of tools from algebraic geometry as pioneered by de Jong and Lieblich. Nevertheless, the standard period-index conjectures are still wide open. I will describe joint work with Antieau, Krashen, Lieblich, and Ingalls on the case of surfaces over a local field, where we establish a universal period-index bound (away from the residual characteristic). Some new ingredients in the proof are the use of Gabber’s recent refined resolution results, the deformation theory of twisted sheaves on reducible surfaces, and an approach to ramification-splitting techniques à la Saltman and Pirutka that uses toroidal geometry.

Qile Chen (Boston College) -- Witten's r-spin class via logarithmic compactification

FJRW theory/Witten's top Chern classes were constructed by Fan-Jarvis-Ruan and Chang-Li-Li. The authors proved that the corresponding virtual cycles are represented by Chow cycles supported on proper loci inside non-proper moduli spaces parameterizing sections of certain line bundles. In this talk, I will focus on the example of r-spin curves to introduce a logarithmic approach using stable logarithmic maps of Abramovich-Chen-Gross-Siebert to construct a proper moduli space. This proper moduli space carries a reduced perfect obstruction theory whose associated virtual cycle is Witten’s r-spin class. Our logarithmic approach can be further applied to many interesting cases including FJRW theory, and more generally hybrid models. This is the first step toward a localization formula for higher genus invariants conjectured by Felix Janda.

This is a joint work in progress with Felix Janda, Yongbin Ruan, Adrien Sauvaget, and Dimitri Zvonkine.

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